Berkovich 和 Dhar 关于钩长的两个猜想的证明

中文摘要 AI 辅助

我们证明了 Berkovich 和 Dhar 关于有界划分中系数不等式和长度为二的钩对的两个猜想的更强形式。系数不等式蕴含了钩对的相应不等式。特别地，对于每个固定大小和最大部分的共同界，奇数划分中此类钩对的总数至少与不同划分中的总数相同。我们通过基于 Glaisher 双射的显式保权注入给出组合证明，并使用有限积恒等式和已知钩生成函数给出解析证明。

英文摘要

We prove stronger forms of two conjectures of Berkovich and Dhar concerning a coefficientwise inequality and pairs of hooks of length two in bounded partitions. The coefficientwise inequality implies the corresponding inequality for hook pairs. In particular, for each fixed size and common bound on the largest part, the total number of such pairs in odd partitions is at least that in distinct partitions. We give combinatorial proofs by explicit weight-preserving injections based on Glaisher's bijection and analytic proofs using finite product identities and known hook generating functions.

发表机构

• Shanghai Normal University（上海师范大学）

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